# eba {eba}

### Description

Fits a (multi-attribute) probabilistic choice model by maximum likelihood.

### Usage

eba(M, A = 1:I, s = rep(1/J, J), constrained = TRUE) OptiPt(M, A = 1:I, s = rep(1/J, J), constrained = TRUE) ## S3 method for class 'eba': summary((object, ...)) ## S3 method for class 'eba': anova((object, ..., test = c("Chisq", "none")))

### Arguments

- M
- a square matrix or a data frame consisting of absolute choice frequencies; row stimuli are chosen over column stimuli
- A
- a list of vectors consisting of the stimulus aspects; the default is
`1:I`

, where`I`

is the number of stimuli - s
- the starting vector with default
`1/J`

for all parameters, where`J`

is the number of parameters - constrained
- logical, if TRUE (default), parameters are constrained to be positive
- object
- an object of class
`eba`

, typically the result of a call to`eba`

- test
- should the p-values of the chi-square distributions be reported?
- ...
- additional arguments; none are used in the summary method; in the anova method they refer to additional objects of class
`eba`

.

### Details

`eba`

is a wrapper function for `OptiPt`

. Both functions can be used interchangeably. See Wickelmaier & Schmid (2004) for further details.

The probabilistic choice models that can be fitted to paired-comparison data are the Bradley-Terry-Luce (BTL) model (Bradley, 1984; Luce, 1959), preference tree (Pretree) models (Tversky & Sattath, 1979), and elimination-by-aspects (EBA) models (Tversky, 1972), the former being special cases of the latter.

`A`

represents the family of aspect sets. It is usually a list of vectors, the first element of each being a number from 1 to `I`

; additional elements specify the aspects shared by several stimuli. `A`

must have as many elements as there are stimuli. When fitting a BTL model, `A`

reduces to `1:I`

(the default), i.e. there is only one aspect per stimulus.

The maximum likelihood estimation of the parameters is carried out by `nlm`

. The Hessian matrix, however, is approximated by `fdHess{nlme}`

. The likelihood functions `L.constrained`

and `L`

are called automatically.

See `group.test`

for details on the likelihood ratio tests reported by `summary.eba`

.

### Values

- coefficients
- a vector of parameter estimates
- estimate
- same as
`coefficients`

- logL.eba
- the log-likelihood of the fitted model
- logL.sat
- the log-likelihood of the saturated (binomial) model
- goodness.of.fit
- the goodness of fit statistic including the likelihood ratio fitted vs. saturated model (-2logL), the degrees of freedom, and the p-value of the corresponding chi-square distribution
- u.scale
- the unnormalized utility scale of the stimuli; each utility scale value is defined as the sum of aspect values (parameters) that characterize a given stimulus
- hessian
- the Hessian matrix of the likelihood function
- cov.p
- the covariance matrix of the model parameters
- chi.alt
- the Pearson chi-square goodness of fit statistic
- fitted
- the fitted paired-comparison matrix
- y1
- the data vector of the upper triangle matrix
- y0
- the data vector of the lower triangle matrix
- n
- the number of observations per pair (
`y1 + y0`

) - mu
- the predicted choice probabilities for the upper triangle
- nobs
- the number of pairs

### References

Bradley, R.A. (1984). Paired comparisons: Some basic procedures and examples. In P.R. Krishnaiah & P.K. Sen (eds.), *Handbook of Statistics, Volume 4*. Amsterdam: Elsevier.

Luce, R.D. (1959). *Individual choice behavior: A theoretical analysis*. New York: Wiley.

Tversky, A. (1972). Elimination by aspects: A theory of choice. *Psychological Review*, **79**, 281--299.

Tversky, A., & Sattath, S. (1979). Preference trees. *Psychological Review*, **86**, 542--573.

Wickelmaier, F., & Schmid, C. (2004). A Matlab function to estimate choice model parameters from paired-comparison data. *Behavior Research Methods, Instruments, and Computers*, **36**, 29--40.

### See Also

`strans`

, `uscale`

, `cov.u`

, `group.test`

, `wald.test`

, `plot.eba`

, `residuals.eba`

, `logLik.eba`

, `simulate.eba`

, `kendall.u`

, `circular`

, `trineq`

, `thurstone`

, `nlm`

.

### Examples

data(celebrities) # absolute choice frequencies btl1 <- eba(celebrities) # fit Bradley-Terry-Luce model A <- list(c(1,10), c(2,10), c(3,10), c(4,11), c(5,11), c(6,11), c(7,12), c(8,12), c(9,12)) # the structure of aspects eba1 <- eba(celebrities, A) # fit elimination-by-aspects model summary(eba1) # goodness of fit plot(eba1) # residuals versus predicted values anova(btl1, eba1) # model comparison based on likelihoods confint(eba1) # confidence intervals for parameters uscale(eba1) # utility scale ci <- 1.96 * sqrt(diag(cov.u(eba1))) # 95% CI for utility scale values dotchart(uscale(eba1), xlim=c(0, .3), main="Choice among celebrities", xlab="Utility scale value (EBA model)", pch=16) # plot the scale arrows(uscale(eba1)-ci, 1:9, uscale(eba1)+ci, 1:9, .05, 90, 3) # error bars abline(v=1/9, lty=2) # indifference line mtext("(Rumelhart & Greeno, 1971)", line=.5)

Documentation reproduced from package eba, version 1.7-1. License: GPL (>= 2)